Tensor product bundle

{{distinguish|text=a tensor bundle, a vector bundle whose section is a tensor field}}

In differential geometry, the tensor product of vector bundles {{mvar|E}}, {{mvar|F}} (over the same space {{mvar|X}}) is a vector bundle, denoted by {{math|E ⊗ F}}, whose fiber over each point {{math|x ∈ X}} is the tensor product of vector spaces {{math|E_x ⊗ F_x}}.To construct a tensor-product bundle over a paracompact base, first note the construction is clear for trivial bundles. For the general case, if the base is compact, choose {{mvar|E{{'}}}} such that {{math|E ⊕ E{{'}}}} is trivial. Choose {{mvar|F{{'}}}} in the same way. Then let {{math|E ⊗ F}} be the subbundle of {{math|(E ⊕ E{{'}}) ⊗ (F ⊕ F{{'}})}} with the desired fibers. Finally, use the approximation argument to handle a non-compact base. See Hatcher for a general direct approach.

Example: If {{mvar|O}} is a trivial line bundle, then {{math|E ⊗ O {{=}} E}} for any {{mvar|E}}.

Example: {{math|E ⊗ E^∗}} is canonically isomorphic to the endomorphism bundle {{math|End(E)}}, where {{math|E^∗}} is the dual bundle of {{mvar|E}}.

Example: A line bundle {{mvar|L}} has a tensor inverse: in fact, {{math|L ⊗ L^∗}} is (isomorphic to) a trivial bundle by the previous example, as {{math|End(L)}} is trivial. Thus, the set of the isomorphism classes of all line bundles on some topological space {{mvar|X}} forms an abelian group called the Picard group of {{mvar|X}}.